ON THE MOTION OF A VISCOUS FLUID 189
where H is a single-valued function, subject to V2ff = 0. If Ł, TJO, & be the rotations,
v0 dv0\ d __ d __0 .
and thus (9) requires that
In two dimensions the dynamical equation reduces to DŁ0/Dt = 0*, so that Ł is constant along a stream-line. Among the cases included are the motion between two planes
and the motion in circles between two coaxal cylinders (Ł = constant). Also, without regard to the form of the boundary, the uniform rotation, as of a solid body, expressed by
^ __ fjy y __ __ Qrg t>> _ ('12')
In all these cases F is an absolute minimum.
Conversely, if the conditions (9) be not satisfied, it will be possible to find a motion for which F< F0. To see this choose a place as origin of coordinates where dV-uQ/dy is not equal to d^\/dx. Within a small sphere described round this point as centre let u' = Gy, v = Gx, w' = 0, and let u' = 0, v' = 0, w' 0 outside the sphere, thus satisfying the prescribed boundary conditions. Then in (2)
f f
(w'V2/Ť0 + v'V\ + w'V'2w0)dxdydz~G\ (y^2ua ,%'V2va)d,xdydz, ...(13) J J
the integration being over the sphere. Within this small region we may take
so that (13) reduces to
Since the sign of G is at disposal, this may be made positive or negative at pleasure. Also I" in (2) may be neglected as of the second order when u ', v', w' are small enough. It follows that F is not an absolute minimum for u0, v0, WQ, unless the conditions (9) are satisfied.
Korteweg has also shown that the slow motion of a viscous fluid denoted. by M0 is stable. "When in a given region occupied by viscous
* Where D/Dt = d/dt'+ u d/dx + v d/dy + iv d/cte. a velocity-potential $, so that u d^/dx, &c.,